Robust convergence result of discontinuous Galerkin stabilization method for two-dimensional reaction–diffusion equation with discontinuous source term

Abstract

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order $\mathcal{O}({\epsilon }^{1/4}{N}^{-1}+N^{-1}\ln N)$. Numerical tests are carried out in support of theoretical findings.

Keywords:

• Robust convergence
• discontinuous Galerkin stabilization
• the NIPG method
• Shishkin mesh

AMS Subject Classifications:

• 65L70
• 65L20
• 65L60

Acknowledgements

The authors wish to thank the anonymous referees for their remarks that contributed to improve the presentation.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.